Linear Algebra and Its Applications · Problem 2.4.36

Vector Spaces — Gilbert Strang

Mathematics · four-fundamental-subspaces

Problem

Without multiplying matrices, find bases for the row and column spaces of

A

A = \begin{bmatrix} 1 & 2 \\ 4 & 5 \\ 2 & 7 \end{bmatrix} \begin{bmatrix} 3 & 0 & 3 \\ 1 & 1 & 2 \end{bmatrix}.

How do you know from these shapes that

A

is not invertible?